The main door of the single-engine Cherokee flew open shortly after take-off.
Ahhhh, no it didn't, it popped open, and trailed a few inches upon, neither opening more, nor being willing to be closed. This should be trained as an acknowledge and ignore event. If everything else is fine, you can play with the rudder, and probably get it closed, but you can also break the door handle if you fight against it.
There is no excuse for loosing control of the airplane. As per FAA AC105-2c, all PA-28-140, 160 & 180s may be approved to fly with the cabin door removed.